Am I obliged to pay the 2,0 Saudi Riyals to that employee if he is unable to recover it, knowing that I only filled in his personal data as a service, and I made a mistake that led to this loss?

You, as a trustee (mu'taman), are not liable for the error you made while filling out the data, as long as you did not fall short or act negligently, because the Prophet, peace and blessings be upon him, said: "There is no liability upon a trustee." If the dispute is not resolved amicably, then the reference point is the Sharia courts.